What is the integral of 4pix(6+x^{3/2}) ?

The integral of 4pix(6+x^{3/2}) is 4pi(3x^2+2/7 x^{7/2})+C

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integral of 4pix(6+x^{3/2})

\int 4 π x (6 + x^{\frac{3}{2}}) d x

4 π (3 x^{2} + \frac{2}{7} x^{\frac{7}{2}}) + C

Solution steps

\int 4 π x (6 + x^{\frac{3}{2}}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 4 π \cdot \int x (6 + x^{\frac{3}{2}}) d x

Expand

x (6 + x^{\frac{3}{2}}) : 6 x + x^{\frac{5}{2}}

= 4 π \cdot \int 6 x + x^{\frac{5}{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 π (\int 6 x d x + \int x^{\frac{5}{2}} d x)

\int 6 x d x = 3 x^{2}

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

= 4 π (3 x^{2} + \frac{2}{7} x^{\frac{7}{2}})

Add a constant to the solution

= 4 π (3 x^{2} + \frac{2}{7} x^{\frac{7}{2}}) + C

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What is the integral of 4pix(6+x^{3/2}) ?
The integral of 4pix(6+x^{3/2}) is 4pi(3x^2+2/7 x^{7/2})+C